Some Extremal Problems On Cycle Length

Let n,r,t be positive integers and let G be a simple,connect and undirected graph of order n.A graph G of order n is said to be r-(p₀,…,p_t-1)-pancyclic if G contains exactly pi?0?i?t-1?cycles Cr+tj+i for r+tj+t-1?n and j is the number of p₀,…,p_t-1 repeats.A graph G of order n is said to be r-(p₀,…,p_t-1)-oddpancyclic?or bipancyclic? if G contains exactly pi?0?i?f-1? cycles Cr+tj+i for r+tj+t-1?n and j is the number of P0,…,p_t-1 repeats.The cycle length distribution of a graph of order n is?c₁,c₂,…,cn?,where ci is the number of cycles of length i,for i=1,2,…,n.Let g?a1,…,an?denote the minimum possible number of edges in a graph which satisfies c_i?a_i where ai is a nonnegative integer.In this paper,we mainly consider some extremal problems on the cycle length of graphs.The following are our main results:1.We provide a construction for r-?p₀,…,p₇?- pancyclic graphs.Similar method are used to construct r-?p₀,…,p₇?- oddpancyclic ?or bipancyclic? graphs. Moreover,we also construct a class of graphs with cycle length distribution ?0,0,c₃,c₄,…,c_n?,where ci?ai,i=3,4,…,n and give the lower bounds for g?0,0,a₃,a₄,…,a_n?,respectively?2.We determine the minimum possible size of simple graphs of order n with at least two t-cycles for each t?{3,…,n},where 3?n?19.